Universal estimators of a vector parameter

Let x be a random sample with a distribution depending on a vector parameter [theta] [set membership, variant] m. The description of distributions and generalized prior densities on m is given, for which the generalized Bayes estimator of [theta], based on x, is the same for all symmetric loss functions. These distributions form a special subclass of exponential family. The specification of this result for the case of a location parameter is considered. The proof of the main theorem is based on the solution of a functional equation of D'Alembert's type.